Prerequisites Almost essential Consumption and Uncertainty Risk MICROECONOMICS Principles and Analysis Frank Cowell November 2006
Risk and uncertainty • In dealing with uncertainty a lot can be done without introducing probability. • Now we introduce a specific probability model • This could be some kind of exogenous mechanism • Could just involve individual’s perceptions • Facilitates discussion of risk • Introduces new way of modelling preferences
Overview... Risk Probability An explicit tool for model building Risk comparisons Special Cases Lotteries
Probability • What type of probability model? • A number of reasonable versions: • Public observable • Public announced • Private objective • Private subjective • Need a way of appropriately representing probabilities in economic models. Lottery government policy? coin flip emerges from structure of preferences.
Ingredients of a probability model • We need to define the support of the distribution • The smallest closed set whose complement has probability zero • Convenient way of specifying what is logically feasible (points in the support) and infeasible (other points). • Distribution function F • Represents probability in a convenient and general way. • Encompass both discrete and continuous distributions. • Discrete distributions can be represented as a vector • Continuous distribution – usually specify density function • Take some particular cases: a collection of examples
Some examples • Begin with two cases of discrete distributions • #W = 2. Probability p of value x0; probability 1–p of value x1. • #W = 5. Probability pi of value xi, i = 0,...,4. • Then a simple example of continuous distribution with bounded support • The rectangular distribution – uniform density over an interval. • Finally an example of continuous distribution with unbounded support
Discrete distribution: Example 1 • Suppose of x0 and x1 are the only possible values. • Below x0 probability is 0. • Probability of x ≤ x0is p. 1 • Probability of x ≥ x0 but less than x1 is p. F(x) • Probability of x ≤ x1 is 1. p x x0 x1
Discrete distribution: Example 2 • There are five possible values: x0 ,…, x4. • Below x0 probability is 0. • Probability of x ≤ x0is p0. 1 • Probability of x ≤ x1 is p0+p1. • Probability of x ≤ x2 is p0+p1 +p2. p0+p1+p2+p3 • Probability of x ≤ x3 is p0+p1+p2+p3. F(x) p0+p1+p2 • Probability of x ≤ x4 is 1. p0+p1 • p0 + p1+ p2+ p3+ p4 = 1 p0 x x2 x3 x4 x0 x1
“Rectangular” : density function • Suppose values are uniformly distributed between x0 and x1. • Below x0 probability is 0. f(x) x x0 x1
Rectangular distribution • Values are uniformly distributed over the interval[x0,x1]. • Below x0 probability is 0. 1 • Probability of x ≥ x0 but less than x1 is [x x0] / [x1 x0] . • Probability of x ≤ x1 is 1. F(x) x x0 x1
Lognormal density • Support is unbounded above. • The density function with parameters m=1, s=0.5 • The mean x 0 1 2 3 4 5 6 7 8 9 10
x 0 1 2 3 4 5 6 7 8 9 10 Lognormal distribution function 1
Overview... Risk Probability Shape of the u-function and attitude to risk Risk comparisons Special Cases Lotteries
Risk aversion and the function u • With a probability model it makes sense to discuss risk attitudes in terms of gambles • Can do this in terms of properties of “felicity” or “cardinal utility” function u • Scale and origin of uare irrelevant • But the curvature of uis important. • We can capture this in more than one way • We will investigate the standard approaches... • ...and then introduce two useful definitions
Risk aversion and choice • Imagine a simple gamble • Two payoffs with known probabilities: • xREDwith probability pRED. • xBLUEwith probability pBLUE. • Expected value Ex = pREDxRED+ pBLUExBLUE. • A “fair gamble”: stake money is exactly Ex . • Would the person accept all fair gambles? • Compare Eu(x) with u(Ex) depends on shape of u
u u(x) Risk-loving u u Risk-neutral Risk-averse u(x) u(x) x x xRED xRED xBLUE xBLUE Ex Ex x xRED xBLUE Ex Attitudes to risk • Shape of u associated with risk attitude • Neutrality: will just accept a fair gamble • Aversion: will reject some better-than-fair gambles • Loving: will accept some unfair gambles
pRED – _____ pBLUE Risk premium and risk aversion • A given income prospect • The certainty equivalent income xBLUE • Slope gives probability ratio • Mean income • The risk premium. _ – • Risk premium: • Amount that amount you would sacrifice to eliminate the risk • Useful additional way of characterising risk attitude • P • P0 example xRED O x Ex
An example... • Two-state model • Subjective probabilities (0.25, 0.75) • Single-commodity payoff in each case
Eu(x) Ex Risk premium: an example u • Utility values of two payoffs • Expected payoff and the utility of expected payoff. • Expected utility and the certainty-equivalent u(Ex) u(x) u(xRED) • The risk premium again u(Ex) • Eu(x) amount you would sacrifice to eliminate the risk u(xBLUE) x xRED xBLUE x Ex x
Change the u-function u • The utility function and risk premium as before • Now let the utility function become “flatter”… u(xRED) • Making the u-function less curved reduces the risk premium… • …and vice versa • More of this later u(xBLUE) u(xBLUE) x xRED x xBLUE x Ex
An index of risk aversion? • Risk aversion associated with shape of u • second derivative • or “curvature” • But could we summarise it in a simple index or measure? • Then we could easily characterise one person as more/less risk averse than another • There is more than one way of doing this
Absolute risk aversion • Definition of absolute risk aversion for scalar payoffs. uxx(x) a(x) := ux(x) • For risk-averse individuals a is positive. • For risk-neutral individuals a is zero. • Definition ensures that a is independent of the scale and the origin of u. • Multiply u by a positive constant… • …add any other constant… • a remains unchanged.
Relative risk aversion • Definition of relative risk aversion for scalar payoffs: uxx(x) r(x) := x ux(x) • Some basic properties of r are similar to those of a: • positive for risk-averse individuals. • zero for risk-neutrality. • independent of the scale and the origin of u • Obvious relation with absolute risk aversion: • r(x) = xa(x)
Concavity and risk aversion • Draw the function u again. u • Change preferences: φ is a concave function of u. • Risk aversion increases. lower risk aversion u(x) utility û(x) More concave uimplies higher risk aversion û =φ(u) higher risk aversion now to the interpretations x payoff
Interpreting a and r • Think of a as a measure of the concavity of u. • Risk premium is approximately ½ a(x) var(x). • Likewise think of r as the elasticity of marginal u. • In both interpretations an increase in the “curvature” of u increases measured risk aversion. • Suppose risk preferences change… • u is replaced by û , where û=φ(u) and φ is strictly concave • Then both a(x) and r(x) increase for all x. • An increase in a or r also associated with increased curvature of IC…
Alf xBLUE xBLUE Alf Charlie Bill xRED xRED O O Another look at indifference curves • Both uand p determine the shape of the IC • Alf and Bill differ in risk aversion • Alf and Charlie differ in subjective probability Same us but different ps Same ps but different us
Overview... Risk Probability CARA and CRRA Risk comparisons Special Cases Lotteries
Special utility functions? • Sometimes convenient to use special assumptions about risk. • Constant ARA • Constant RRA • By definition r(x) = xa(x) • Differentiate w.r.t. x: dr(x) da(x) = a(x) + x dx dx • So one could have, for example: • constant ARA and increasing RRA • constant RRA and decreasing ARA • or, of course, decreasing ARA and increasing RRA
Special case 1: CARA • We take a special case of risk preferences • Assume that a(x) = a for all x • Felicity function must take the form 1 u(x) := eax a • Constant Absolute Risk Aversion • This induces a distinctive pattern of indifference curves...
Constant Absolute RA • Case where a= ½ xBLUE • Slope of IC the same along 45° ray (standard vNM) • But, for CARA, slope of IC the same along any45° line xRED O
CARA: changing a • Case where a = ½ (as before) xBLUE • Change ARA to a = 2 xRED O
Special case 2: CRRA • Another important special case of risk preferences. • Assume that r(x) = r for all x. • Felicity function must take the form 1 u(x) := x1 r 1 r • Constant Relative Risk Aversion • Again induces a distinctive pattern of indifference curves...
Constant Relative RA • Case where r= 2 xBLUE • Slope of IC the same along 45° ray (standard vNM) • For CRRA, slope of IC is the same along any ray through O • ICs are homothetic. xRED O
CRRA: changing r • Case where r= 2 (as before) xBLUE • Change RRA to r= ½ xRED O
CRRA: changing p • Case where r= 2 (as before) xBLUE • Increase probability of state RED xRED O
Overview... Risk Probability Probability distributions as objects of choice Risk comparisons Special Cases Lotteries
Lotteries • Consider lottery as a particular type of uncertain prospect. • Take an explicit probability model. • Assume a finite number of states-of-the-world • Associated with each state w are: • A known payoff xw , • A known probabilitypw ≥ 0. • The lottery is the probability distribution over the “prizes” xw, w=1,2,...,. • The probability distribution is just the vector p:= (p1,,p2 ,…,,p) • Of course, p1+ p2 +…+p = 1. • What of preferences?
The probability diagram: #W=2 • Probability of state RED • Probability of stateBLUE pBLUE • Cases of perfect certainty. • Cases where 0 < p< 1 • The case (0.75, 0.25) • (0,1) pRED+pBLUE= 1 • Only points on the purple line make sense. • This is an 1-dimensional example of a simplex • (0, 0.25) (0.75, 0) pRED • (1,0)
The probability diagram: #W=3 pBLUE • Third axis corresponds to probability of state GREEN • There are now three cases of perfect certainty. • (0,0,1) • Cases where 0 < p< 1 • The case (0.5, 0.25, 0.25) pRED + pGREEN + pBLUE= 1 pGREEN • Only points on the purple triangle make sense, • This is a 2-dimensional example of a simplex • • (0,1,0) (0, 0, 0.25) (0, 0.25 , 0) 0 • (1,0,0) (0.5, 0, 0) pRED
Probability diagram #W=3 (contd.) • All the essential information is in the simplex • (0,0,1) • Display as a plane diagram • The equi-probable case • The case (0.5, 0.25, 0.25) •(1/3,1/3,1/3) •(0.5, 0.25, 0.25) • (0,1,0) • . (1,0,0)
Preferences over lotteries • Take the probability distributions as objects of choice • Imagine a set of lotteries p°, p', p",... • Each lottery p has same payoff structure • State-of-the-world w has payoff xw • ... and probabilitypw° or pw' or pw" ... depending on which lottery • We need an alternative axiomatisation for choice amongst lotteries p
Axioms on preferences • Transitivity over lotteries • If p°<p' and p'<p" ... • ...then p°<p". • Independence of lotteries • If p°<p' and l(0,1)... • ...then lp° + [1-l]p" <lp' + [1-l]p" • Continuity over lotteries • If p°Âp'Âp" then there are numbers l and m such that • lp° +[1-l]p" Â p' • p' Â mp° +[1-m]p"
Basic result • Take the axioms transitivity, independence, continuity • Imply that preferences must be representable in the form of a von Neumann-Morgenstern utility function: åpw u(xw) w ÎW • or equivalently: åpw uw w ÎW whereuw := u(xw) • So we can also see the EU model as a weighted sum of ps.
p-indifference curves • Indifference curves over probabilities. • (0,0,1) • Effect of an increase in the size of uBLUE • (0,1,0) • . (1,0,0)
What next? • Simple trading model under uncertainty • Consumer choice under uncertainty • Models of asset holding • Models of insurance • This is in the presentation Risk Taking